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c. The reflexive closure of a quasi order is a partial order.

d. Every finite poset has a minimal element and a maximal element

10. List the ordered pairs in the relation R from A = {0,1, 2, 3} to B = {0, 1, 2, 3, 4} where (a, b) R if and only if

a. a>b.

b. a + b = 3.

c. a divides b.

d. a - b = 0.

e. gcd( a , b ) = 1.

f. lcm( a , b ) = 6.

11. Recursively define the relation { ( a , b ) | a=2b }, where a and b are natural numbers.

12. List unary relation on {1, 2, 3}.

13. Prove that there are 2 n2 binary relations on a set of cardinality n .

14. For each of the following relations on the set {1, 2, 3, 4}, decide whether it is reflexive, symmetric, antisymmetric and/or transitive.

a. {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)}

b. {(1, 3), (1, 4), (2, 3), (3, 4)}

c. {(1, 1), (1, 3), (1, 4), (2, 1), (2, 3), (2, 4), (3, 1), (3, 3), (3, 4)}

15. Determine whether the relation R on the set of all integers is reflexive, symmetric, antisymmetric, and/or transitive, where ( x , y ) ∈ R if and only if

a. x is divisible by y .

b. x y.

c. y = x + 2 or y = x - 2.

d. x = y ² + 1 .

16. Let A be the set of people in your town. Let R 1 be the unary relation representing the people in your town who were registered in the last election  and R 2 be the unary relation representing the people in your town who voted in the last election.  Describe the 1-tuples in each of the following relations.

a. R 1 ∪ R 2.

b. R 1 ∩ R 2.

17. Draw the directed graph that represents the relation {( a , b ), ( a , c ), ( b , c ), ( c , b ), ( c , c ), ( c , d ), ( d , a ), ( d , b )}.

18. Let R be the parent-child relation on the set of people that is, R = { ( a, b ) | a is a parent of b }.  Let S be the sibling relation on the set of people that is, R = { ( a , b ) | a and b are siblings (brothers or sisters) }.  What are S o R and R o S ?

19. Let R be a reflexive relation on a set A .  Show that R n is reflexive for all positive integers n .

20. Let R be the relation on the set { 1, 2, 3, 4} containing the ordered pairs (1, 1), (1, 2), (2, 2), (2, 4), (3, 4), and (4, 1).  Find

a. the reflexive closure of R

b. symmetric closure of R   and

c. transitive closure of R .

21. Let R be the relation { ( a, b ) | a is a (integer) multiple of b } on the set of integers.  What is the symmetric closure of R ?

22. Suppose that a binary relation R on a set A is reflexive.  Show that   R*   is reflexive, where   R* = i = 1 n R i size 12{ union rSub { size 8{i=1} } rSup { size 8{n} } {R} rSup { size 8{i} } } {} .

23. Which of the following relations on {1, 2, 3, 4} are equivalence relations?  Determine the properties of an equivalence relation that the others lack.

a. {(1, 1), (2, 2), (3, 3), (4, 4)}

b. {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)}

c. {(1, 1), (1, 2), (1, 4), (2, 2), (2, 4), (3, 3), (4, 1), (4, 2), (4, 4)}

24. Suppose that A is a nonempty set, and f is a function that has A as its domain.  Let R be the relation on A consisting of all ordered pairs ( x , y ) where f(x) = f(y) .

a. Show that R is an equivalence relation on A .

b. What are the equivalence classes of R ?

25. Show that propositional equivalence is an equivalence relation on the set of all compound propositions.

26. Give a description of each of the congruence classes modulo 6.

27. Which of the following collections of subsets are partitions of {1, 2, 3, 4, 5, 6}?

a. {1, 2, 3}, {3, 4}, {4, 5, 6}

b. {1, 2, 6}, {3, 5}, {4}

c. {2, 4, 6}, {1, 5}

d. {1, 4, 5}, {2, 3, 6}

28. Consider the equivalence relation on the set of integers R = { ( x, y ) | x - y is an integer}.

a. What is the equivalence class of 1 for this equivalence relation?

b. What is the equivalence class of 0.3 for this equivalence relation?

29. Which of the following are posets?

a. ( Z,  =  )

b. ( Z, ≠)

c. ( A collection of sets, ⊆).

30. Draw the Hasse diagram for the divisibility relation on the following sets

a. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

b. {1, 2, 5, 8, 16, 32}

31. Answer the following questions concerning the poset ({{1}, {2}, {3}, {4}, {1, 3}, {1, 4}, {2, 4}, {3, 4}, {1, 2, 4}, {2, 3, 4}},⊆).

a. Find the maximal elements.

b. Find the minimal elements.

c. Is there a greatest element?

d Is there a least element?

e. Find all upper bounds of {{2}, {4}}.

f. Find the least upper bound of {{2}, {4}}, if it exists.

g. Find all lower bounds of {{1, 2, 4}, {2, 3, 4}}

h. Find the greatest lower bound of {{1, 2, 4}, {2, 3, 4}}, if it exists.

Questions & Answers

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Eke Reply
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Joseph
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Maira Reply
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Ali
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learn
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Google
da
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Bhagvanji
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Giriraj
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da
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narayan
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RAW Reply
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Damian
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
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Brian Reply
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
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LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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LITNING
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Sahil
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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Mahi
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Rafiq
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Anam
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Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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Bob Reply
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Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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Source:  OpenStax, Discrete structures. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10768/1.1
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